Normal Distribution

What is Normal Distribution in Statistics?

Normal Distribution is a bell-shaped frequency distribution curve which helps describe all the possible values a random variable can take within a given range with most of the distribution area is in the middle and few are in the tails, at the extremes. This distribution has two key parameters: the mean (µ) and the standard deviation (σ) which plays key role in assets return calculation and in risk management strategy.

How to Interpret Normal Distribution

Graph1

The above figure shows that the statistical normal distribution is a bell-shaped curve. The range of possible outcomes of this distribution is the whole real numbers lying between -∞ to +∞. The tails of the bell curve extend on both sides of the chart (+/-) without limits.

  • Approximately 68% of all observation fall within +/- one standard deviation(σ)
  • Approximately 95% of all observation fall within +/- two standard deviations (σ)
  • Approximately 99% of all observation fall within +/- three standard deviations (σ)

It has a skewness of zero (symmetry of a distribution). If the distribution of data is asymmetric, then the distribution is uneven if the data set has skewness greater than zero or positive skewness. Then, the right tail of the distribution is more prolonged than the left, and for negative skewness (less than zero) left tail will be longer than the right tail.

It has a kurtosis of 3 (measures peakedness of a distribution), which indicates distribution is neither too peaked nor too thin tails. If the kurtosis is more than three than distribution is more peaked with fatter tails, and if the kurtosis is less than three, then it has thin tails, and the peak point is lower than the normal distribution.

Characteristics

  • They represent a family of distribution where mean & deviation determine the shape of the distribution.
  • The mean, median, and mode of this distribution are all equal.
  • Half of the values are to the left of the center and the other half to the right.
  • The total value under the standard curve will always be one.
  • Most likely, distribution is at the center, and fewer values lie in the tail end.

Normal-Distribution

Transformation (Z)

The Probability density function(PDF) of a random variable (X) following distribution is given by:

Normal Distribution (Formula)

where -∞ < x < ∞ ; -∞ < µ < ∞ ; σ > 0

Where,

  •  F(x) = Normal probability Function
  •  x = Random variable
  • µ = Mean of distribution
  • σ = Standard deviation of the distribution
  • π = 3.14159
  • e = 2.71828

Transformation Formula

Transformation Formula

Where,

  • X = Random variable

Examples of Normal Distribution in Statistics

Let’s discuss the following examples.

Example #1

Suppose a company has 10000 employees and multiple salaries structure as per the job role in which employee works. The salaries are generally distributed with the population mean of µ = $60,000, and the population standard deviation σ = $15000. What will be the probability that randomly selected employee has a salary less than $45000 annually.

Graph 2

Solution

As shown in the above figure, to answer this question, we need to find out the area under the normal curve from 45 to the left side tail. Also, we need to use Z- table value to get the right answer.

Firstly, we need to convert the given mean and standard deviation into a standard normal distribution with mean (µ)= 0 and standard deviation (σ) =1 using the transformation formula.

After the conversion, we need to look up the Z- table to find out the corresponding value, which will give us the correct answer.

Given,

  • Mean (µ) = $60,000
  • Standard deviation (σ) = $15000
  • Random Variable (x) = $45000

Transformation (z) = (45000 – 60000 / 15000)

Transformation (z) = -1

Now the value that is equivalent to -1 in Z-table is 0.1587, which represents the area under the curve from 45 to the way to left. It indicated that when we randomly select an employee, the probability of making less than $45000 a year is 15.87%.

Example #2

Now keeping the same scenario as above, find out the probability that randomly selected employee earns more than $80,000 a year using the normal distribution.

Normal distribution formula

Solution

So in this question, we need to find out the shaded area from 80 to right tail using the same formula.

Given,

  • Mean (µ) = $60,000
  • Standard deviation (σ) = $15000
  • Random Variable (X) = $80,000

Transformation (z) = (80000 – 60000 /15000)

Transformation (z) = 1.33

As per the Z-table, the equivalent value of 1.33 is 0.9082 or 90.82%, which shows that the probability of randomly selecting employees earning less than $80,000 annually is 90.82%.

But as per the question, we need to determine the probability of the random employees earning more than $80,000 a year, so we need to subtract value from 100.

  • Random Variable (X) = 100% – 90.82%
  • Random Variable (X) = 9.18%

So the probability that employees earn more than $80,000 per year is 9.18%.

Uses

  • The stock market technical chart is often a bell curve, allowing analysts and investors to make statistical inferences about expected return and risk of stocks.
  • It is used in the real-world, like to determine the most probable best time taken by pizza companies to deliver pizza and many more real applications.
  • Used in comparing heights of a given population set in which most people will have an average size with very few people having above average or below average height.
  • They are used in determining the average academic performance of students, which helps to compare the rank of students.

Conclusion

Normal distribution finds applications in data science and data analytics. Advanced technologies like Artificial Intelligence and machine learning used along with this distribution can give better data quality, which will help individuals and companies in effective decision making.

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This has been a guide to what is Normal Distribution in Statistics and its definition. Here we discuss examples of normal distribution along with its characteristics and uses. You can learn more about financing from the following articles –

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